% This file was created with JabRef 2.5.
% Encoding: GBK

@ARTICLE{Akahori1995,
  author = {Akahori, Jir\^o},
  title = {{Some formulae for a new type of path-dependent option.}},
  journal = {Ann. Appl. Probab. },
  year = {1995},
  volume = {5},
  pages = {383-388},
  number = {2},
  abstract = {{Summary: We present an explicit form of the distribution function
	of the occupation time of a Brownian motion with a constant drift
	(if there is no drift, this is the well-known arc-sine law). We also
	define the $\alpha$-percentile of the stock price and give an explicit
	form of the distribution function of this random variables. Using
	this explicit distribution, we calculate the price of a new type
	of path-dependent option, called the $\alpha$-percentile option.
	This option was first introduced by Miura and is based on order statistics.}},
  classmath = {{*91B24 (Price theory and market structure) 60H30 (Appl. of stochastic
	analysis) 60G44 (Martingales with continuous parameter) }},
  doi = {10.1214/aoap/1177004769},
  file = {:D\:\\eBooks\\papers\\probility\\Jiro Akahori,Some formulae for a new type of path-dependent option.PDF:PDF},
  keywords = {{$\alpha$-percentile of the stock price; $\alpha$-percentile option;
	order statistics}},
  language = {English}
}

@ARTICLE{L.Chaumont04011999,
  author = {Chaumont, L.},
  title = {{A Path Transformation and its Applications to Fluctuation Theory}},
  journal = {J. London Math. Soc.},
  year = {1999},
  volume = {59},
  pages = {729-741},
  number = {2},
  abstract = {We first establish a combinatorial result on deterministic real chains.
	This is then applied to prove a path transformation for chains with
	exchangeable increments. From this transformation we derive an identity
	on order statistics due to Port, together with some extensions. Then
	we give an interpretation of these results in continuous time. We
	extend some identities involving quantiles and occupation times for
	processes with exchangeable increments. In particular, this yields
	an extension of the uniform law for bridges obtained by Knight.},
  doi = {10.1112/S0024610798006929},
  eprint = {http://jlms.oxfordjournals.org/cgi/reprint/59/2/729.pdf},
  file = {:D\:\\eBooks\\papers\\probility\\L. Chaumont, a path transformation and its applications to fluctuation theory.PDF:PDF},
  url = {http://jlms.oxfordjournals.org/cgi/content/abstract/59/2/729}
}

@ARTICLE{Dassios2005,
  author = {Dassios, Angelos},
  title = {{On the quantiles of Brownian motion and their hitting times.}},
  journal = {Bernoulli },
  year = {2005},
  volume = {11},
  pages = {29-36},
  number = {1},
  abstract = {{Summary: The distribution of the $\alpha$-quantile of a Brownian
	motion on an interval $[0,t]$ has been obtained motivated by a problem
	in financial mathematics. We generalize these results by calculating
	an explicit expression for the joint density of the $\alpha$-quantile
	of a standard Brownian motion, its first and last hitting times and
	the value of the process at time $t$. Our results can easily be generalized
	to a Brownian motion with drift. It is shown that the first and last
	hitting times follow a transformed arcsine law.}},
  classmath = {{*60J65 (Brownian motion) }},
  doi = {10.3150/bj/1110228240},
  file = {:D\:\\eBooks\\papers\\probility\\Angelos Dassios, On the quantiles of brownain motion and their hitting times.pdf:PDF},
  keywords = {{arcsine law; hitting times; quantiles of Brownian motion}},
  language = {English}
}

@ARTICLE{dassios1995,
  author = {Dassios, Angelos},
  title = {{The distribution of the quantile of a Brownian motion with drift
	and the pricing of related path-dependent options.}},
  journal = {Ann. Appl. Probab. },
  year = {1995},
  volume = {5},
  pages = {389-398},
  number = {2},
  abstract = {{Let $X$ be Brownian motion with a constant drift and $M(\alpha, t)$
	the $\alpha$-quantile of $X$ up to time $t$, i.e., the smallest value
	$x$ for which the time spent by $X$ below $x$ before $t$ exceeds
	$\alpha t$. The main result of the paper is that $M(\alpha, t)$ has
	the same distribution as the sum of the maximum of $X$ up to time
	$\alpha t$ and the minimum of an independent copy of $X$ up to time
	$(1- \alpha) t$. This is proved analytically by computing the Laplace
	transform of $M(\alpha, t)$ with the help of the Feynman-Kac formula.
	The same identity in law has recently been proved directly by {\it
	P. Embrechts}, {\it L. C. G. Rogers} and {\it M. Yor} [ibid. 5, No.
	3, 757-767 (1995)], using more probabilistic arguments. As an application
	of the basic result, the author recovers a formula by Miura and Akahori
	for the price of a path- dependent financial option.}},
  classmath = {{*60J65 (Brownian motion) }},
  doi = {10.1214/aoap/1177004770},
  file = {:D\:\\eBooks\\papers\\probility\\Angelos Dassios, the distribution of the quantile of a brownian motion with drift and the price of related path-dependent options.PDF:PDF},
  keywords = {{quantiles of Brownian motion with drift; path-dependent options;
	Brownian motion; Feynman-Kac formula}},
  language = {English},
  reviewer = {{M.Schweizer (Berlin)}}
}

@ARTICLE{EmRoge1995,
  author = {Embrechts, P. and Rogers, L.C.G. and Yor, M.},
  title = {{A proof of Dassios' representation of the $\alpha$-quantile of Brownian
	motion with drift.}},
  journal = {Ann. Appl. Probab. },
  year = {1995},
  volume = {5},
  pages = {757-767},
  number = {3},
  abstract = {{Let $X$ be a real-valued Brownian motion with drift. Define for $0
	\leq \alpha \leq 1$ $$M(\alpha, t) = \inf \Biggl\{ x : \int^1_0 1_{\{X_s
	\leq x\}} ds &gt; \alpha t \Biggr\},$$ the $\alpha$-quantile of the
	occupation measure of $X$ on the time interval $[0,t]$. Motivated
	by questions in mathematical finance on the pricing of options, {\it
	A. Dassios} [ibid. 5, No. 2, 389-398 (1995; Zbl 0837.60076)] observed
	the following striking identity in law: $$M(\alpha, t) \overset (d)
	\to = \sup_{0 \leq s \leq \alpha t} X_s + \inf_{0 \leq s \leq (1
	- \alpha)t} X_s', \tag *$$ where $X'$ is an independent copy of $X$.
	As a matter of fact, a discrete time version of this identity has
	been previously proven by {\it J. G. Wendel} [Ann. Math. Stat. 31,
	1034-1044 (1960; Zbl 0118.33701)] and {\it S. C. Port} [J. Math.
	Anal. Appl. 6, 109-151 (1963; Zbl 0114.34101)] in their study of
	ordered statistics of partial sums. The paper under review contains
	two proofs of the identity (*). The first relies on the celebrated
	identity of Sparre-Andersen, and the second on a path decomposition
	of Brownian motion with drift which extends an earlier result of
	the reviewer [in: S\'eminaire de probabilit\'es XXV, Lect. Notes
	Math. 1485, 330-344 (1991; Zbl 0741.60077)]. More recently, the latter
	approach has been extended by {\it Chaumont, M. Yor} and the reviewer
	[``Two chain-transformations and their applications to quantiles''
	(to appear in J. Appl. Probab.)] to explain a more general version
	of (*).}},
  classmath = {{*60J99 (Markov processes) 60J20 (Appl. of discrete Markov processes)
	}},
  doi = {10.1214/aoap/1177004704},
  file = {:D\:\\eBooks\\papers\\probility\\P. Embrechts, L. C. G. Rogers, M. Yor, A Proof of Dassios' Representation of the alpha-Quantile of Brownian Motion with Drift.pdf:PDF},
  keywords = {{alpha-quantile; Brownian motion with drift; path decomposition of
	Brownian motion with drift}},
  language = {English},
  reviewer = {{J.Bertoin (Paris)}}
}

